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Efficient Diversification I

Efficient Diversification I. Covariance and Portfolio Risk Mean-variance Frontier Efficient Portfolio Frontier. Some Empirical Evidence. In 2000, 40% of stocks in Russell 3000 had returns of -20% or worse. Meanwhile, less than 12% of U.S. stock mutual funds had returns of -20% or below.

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Efficient Diversification I

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  1. Efficient Diversification I Covariance and Portfolio Risk Mean-variance Frontier Efficient Portfolio Frontier

  2. Some Empirical Evidence • In 2000, 40% of stocks in Russell 3000 had returns of -20% or worse. • Meanwhile, less than 12% of U.S. stock mutual funds had returns of -20% or below. • Of the 2,397 U.S. stocks in existence throughout 1990s, 22% had negative returns. • In contrast, 0.4% of U.S. equity mutual funds had negative returns.

  3. Diversification and Portfolio Risk • “Don’t put all your eggs in one basket” • Effect of portfolio diversification  Diversifiable risk, non-systematic risk, firm-specific risk, idiosyncratic risk Non-diversifiable risk, systematic risk, market risk 5 15 10 20 # of securities in the portfolio

  4. Covariance and Correlation • Covariance and correlation • Degree of co-movement of two stocks • Covariance: non-standardized measure • Correlation coefficient: standardized measure r2 r2 r2 r1 r1 r1 0<12 <1 -1<12 <0 12 =0

  5. Covariance and Correlation • Example: Two risky assets • Calculating the covariance Means Std. Dev. Cov. Corr.

  6. Diversification and Portfolio Risk • A portfolio of two risky assets • w1: % invested in bond • w2: % invested in stock • Expected return • Variance

  7. Diversification and Portfolio Risk • Example: Portfolio of two risky securities • w in security 1, (1 – w) in security 2 • Expected return (Mean): • Variance • What happens when w changes? • Expected return decreases with increasing w • How about variance ?..

  8. Mean-Variance Frontier • w from 0 to 1 GMVP: Global Minimum Variance Portfolio Mean-variance frontier Security 2 GMVP Security 1

  9. Mean-Variance Frontier • Global Minimum Variance Port. (GMVP) • A unique w • Associated characteristics

  10. Efficient Portfolio Frontier • 67% in Security 1 and 33% in Security 2, what’s so special? • Efficient portfolio has < 67% in 1, and > 33% in 2 w1=0 P Efficient Frontier w1 = .6733 GMVP Inefficient Frontier w1=1

  11. Efficient Portfolio Frontier • Portfolio “P” dominates Security 1 • The same standard deviation • The higher expected return • How to find it? • Since the portfolio has the same standard deviation as Security 1 • Solve the quadratic equation • w = 1 (Security 1) or w = .3465 (Portfolio P)

  12. Efficient Portfolio Frontier • The effect of correlation • Lower correlation means greater risk reduction • Ifr= +1.0, no risk reduction is possible

  13. Efficient Portfolio Frontier • Efficient Portfolio of Many securities • E[rp]: Weighted average of n securities • p2: Combination of all pair-wise covariance measures • Construction of the efficient frontier is complicated • Analytical solution without short-sale constraints • Numerical solution with short-sale constraints • General Features • Optimal combination results in lowest risk for given return • Efficient frontier describes optimal trade-off • Portfolios on efficient frontier are dominant

  14. Efficient Frontier E[r] Efficient frontier Individual assets Global minimum variance portfolio Minimum variance frontier St. Dev.

  15. Wrap-up • How to estimate portfolio return and risk? • What is the mean-variance frontier? • What is the efficient portfolio frontier? • Why do portfolios on efficient frontier dominate other combinations?

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